Doped $t-J$ model on a triangular lattice: Possible application to ${\mathrm{Na}}_x{\mathrm{CoO}}_2\cdot y{\mathrm{H}}_2\mathrm{O}$ and ${\mathrm{Na}}_{1-x}{\mathrm{TiO}}_2$

We report the finding of time-reversal-symmetry-breaking $d_{x^2-y^2}{+id}_{\mathrm{xy}}$ superconducting ground state in the slave-boson mean-field theory for the $t-J$ model on triangular lattice. For $t/J=-5(t/J=-9)$ pairing exists for $x<\mathrm{}13\%$ $\mathrm{}(x<\mathrm{}8\%)\mathrm{}$ upon electron doping, and $x<\mathrm{}56\%$ $\mathrm{}(x<\mathrm{}13\%)\mathrm{}$ upon hole doping. These results are potentially relevant to doped Mott insulators ${\mathrm{Na}}_x{\mathrm{CoO}}_2\cdot y{\mathrm{H}}_2\mathrm{O}$ and ${\mathrm{Na}}_{1-x}{\mathrm{TiO}}_2.$